Comparing AIC of a model and its logit -transformed version

This is not actually a mixed-model question -- it would be better on 
stats.stackexchange.com -- but here we go:

  A clear description of the calculation is here:

https://stats.stackexchange.com/a/100671/2126

  We need -2 * sum(log( d( log(x/(1-x)), "x" )))

Being super-lazy and using sympy

from sympy import *
x = Symbol("x")
simplify(diff(log(x/(1-x)), x))
## -1/(x*(x-1)) = 1/(x*(1-x))

taking -2*log() of this we get

2*sum(log(x*(1-x)))


seedrates <- data.frame(rate = c(50, 75, 100, 125, 150),
                            grain = c(21.2, 19.9, 19.2, 18.4, 17.9)) |>
    transform(pgrain = grain/100, logit_grain = qlogis(grain/100))

m0 <- lm(pgrain~1, data=seedrates)
m1 <- lm(logit_grain ~ 1, data = seedrates)

AIC(m0)
with(seedrates, AIC(m1) + 2*sum(log(pgrain*(1-pgrain))))

These are actually slightly different because of Jensen's inequality 
(the predicted values mean(pgrain) and plogis(mean(logit_grain)) are not 
quite the same), but close enough that I think the computation is done 
correctly.

Hello All,
I would like to compare a model to its logit transformation using the AIC
values.

As a toy example
library(car)

seedrates <- data.frame(rate = c(50, 75, 100, 125, 150),
                         grain = c(21.2, 19.9, 19.2, 18.4, 17.9))
lm <- lm(grain~rate, data=seedrates)
logit.lm <- lm(logit(grain)~rate,data=seedrates)

AIC(lm, logit.lm )

In order to compare these two models using AIC we need to take into account
the JAcobian of the logit transformation.


Here
https://stats.stackexchange.com/questions/61332/comparing-aic-of-a-model-and-its-log-transformed-version

Prof. Ben Bolker mentioned how we can adjust the AIC in the presence of log
transformation by the multiplication of the likelihood by the corresponding
Jacobian to the AIC ... for the case of log{y(n)+1}, it is ?2 ??log{y(n)+1}

I was wondering in the case of logit transformation can I adjust the AIC by
multiplying the likelihood by logit{y(n)+1}.

Any help is greatly appreciated.

Kind regards,

Boby  Mathew